By John Piper - desiringgod.org
Jesus said, "I am the way, and the truth, and the life. No one comes to the Father except through me" (John 14:6). Jesus himself is saying there isn't any other pathway to the Father except through me. Peter, later in the book of Acts, picks that up when he says, "There is salvation in no one else, for there is no other name under heaven given among men by which we must be saved" (Acts 4:12). Peter is taking Jesus' words and carrying them forward into the mission of the church by saying there isn't any other name you can call upon in order to be saved.
Another amazing way Jesus addresses this is that he makes the embrace of himself as the crucified and risen Messiah the litmus test of other religions. If you are talking with somebody who has their set of religious beliefs and you want to test to see whether they have saving faith, put Jesus and his fullness out on the table for them and see if they will embrace him. Jesus said, "You know neither me nor my Father. If you knew me, you would know my Father also" (John 8:19). In other words, knowing Jesus is the condition of truly knowing the Father, that is God, the creator of the world. Jesus himself is saying there isn't any other pathway to the Father except through him.
In John 5:23 Jesus said, "Whoever does not honor the Son does not honor the Father who sent me." He makes himself, the honoring of himself, the litmus test of honoring the Father. Or John 6:45. "Everyone who has heard and learned from the Father comes to me." If a person won't come to Jesus as the Savior, as the one who died for sins, and as the one who rose from the dead, then Jesus says all their claims notwithstanding, they haven't learned from the Father.
Let's go to Paul in the New Testament. One of the most amazing passages that has had missionary impact on me is Romans 5:12 and following. There Paul sets up Adam as the head of our race of people " all of us who have fallen and are under condemnation. Christ is the second Adam " the head of a new humanity " so that in Christ we are now counted as righteous the way we were counted as sinners in Adam.
In Paul's mind, the problem of the entire human race, every religion in the world, and every tribe and people in the world is the same, namely, we are alienated from God in Adam. Jesus comes into the world as the answer to that problem. There isn't another answer out there because there isn't another problem out there. So Paul's conception of how to think about being remedied from our human condition is Adam brought it into being and Christ is the only answer to it.
Then Paul spells it out for missions in Romans 10:14"15 where he says, "How then will they call on him in whom they have not believed? And how are they to believe in him of whom they have never heard? And how are they to hear without someone preaching? And how are they to preach unless they are sent?" Then in Romans 10:17 he clarifies, "Faith comes from hearing, and hearing through the word of Christ."
I think that sequence is Paul"s decisive word about missions in regard to "are there other ways out there for people to call upon the name of the Lord and be saved?" He said, "No, there is one name; It is Jesus. They have to hear about him. In order to hear about him, there must be messengers, missionaries, or preachers to share the gospel." That is the way faith is awakened. And salvation comes through faith.
My conclusion is: Yes, the Bible does present Jesus as the one way among all the religions of the world, which is why there is such a weighty mandate on the church to spread the gospel to all the peoples of the world. There is no salvation without knowing, embracing, and believing Christ as the one who died for our sins and rose again.